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Handbook_of_grid_generation.pdf
Handbook of Grid Generation
Foreword
Contributors
Acknowledgments
Preface: An Elementary Introduction
P-1Discretizations
P-1.1Point Discretization
P-1.2Cell Discretization
P-2Curvilinear (Structured) Grids
P-2.1Boundary-Fitted Grids
P-2.2Block Structure (The Sponge Analogy)
2.3Grid Generation Approaches
P-2.4Variations
P-2.5Transformation
P-3Unstructured Grids
P-3.1Connectivities and Data Structures
P-3.2Grid Generation Approaches
P-3.2.1Triangle and Tetrahedra Creation by Delaunay Triangulation
P-3.2.2Triangle and Tetrahedra Creation by the Advancing Front Method
P-3.2.3Unstructured Grids of Quadrilaterals and Hexahedra
P-3.2.4Surface Mesh Generation
P-3.3Grid Adaptation Techniques
P-3.3.1Grid Refinement
P-3.3.2Grid Movement
P-3.3.3Combinations of Node Movement, Point Enrichment and Derefinement
P-3.3.4Grid Remeshing
Contents
Part I: Block-Structured Grids
Introduction to Structured Grids
Chapter 1: Fundamental Concepts and Approaches
1.1 Introduction
1.2 Mesh Generation Considerations
1.3 Structured Grids
1.3.1 Composite Grids
1.3.1.1 Terminology
1.3.1.2 Forms
1.3.2 Block-Structured Grids
1.3.3 Elliptic Systems
1.3.3.1 Generation System
1.3.3.2 Control Functions
1.3.3.3 Boundary Orthogonality
1.3.3.4 Surface Grids
1.3.4 Hyperbolic System
1.3.5 Algebraic System
1.3.5.1 Transfinite Interpolation
1.3.6 Adaptive Grid Schemes
1.3.6.1 Redistribution of a Fixed Number of Points
1.3.6.2 Local Refinement of a Fixed Set of Points
1.3.6.3 Local Increases in Algorithm Order
1.3.6.4 Formulations
1.4 Unstructured Grids
1.4.1 The Delaunay Triangulation
1.4.1.1 DelaunayVorono Geometrical Construction
1.4.1.2 Algorithm to Construct the Delaunay Triangulation
1.4.2 Point Creation
1.4.2.1 Points Created by an Independent Generation Technique
1.4.2.2 Points Created by Grid Superposition and Successive Subdivision
1.4.2.3 Point Creation Driven by the Boundary Point Distribution
1.4.2.4 Point Creation Controlled by Point and Line Sources
1.4.2.5 Point Creation Controlled by a Background Mesh
1.4.3 Other Unstructured Grid Techniques
1.4.3.1 Advancing Front Methods
1.4.3.2 Quadtree and Octree
1.4.3.3 Hybrid Grids
1.4.4 Unstructured Grid Generation on Surfaces
1.4.5 Adaptation on Unstructured Grids
1.4.5.1 Point Enrichment
1.4.5.2 Node Movement
1.4.5.3 Remeshing
1.4.6 Summary
References
Chapter 2: Mathematics of Space and Surface Grid Generation
2.1 Introduction
2.2 A Rsum of Differential Operations in Curvilinear Coordinates
2.2.1 Representations in Terms of i and i
2.2.2 Differential Operations
2.2.3 Metric Tensor and the Line Element
2.2.4 Differentiation of the Base Vectors
2.2.5 Covariant and Intrinsic Derivatives
2.2.6 Laplacian of a Scalar
2.3 Theory of Curves
2.3.1 A Collection of Usable Formulae for Curves
2.4 Geometrical Elements of the Surface Theory
2.4.1 The Surface Christoffel Symbols
2.4.2 Normal Curvature and the Second Fundamental Form
2.4.3 Principal Normal Curvatures
2.4.4 Mean and Gaussian Curvatures
2.4.5 Derivatives of the Surface Normal; Formulae of Weingarten
2.4.6 Formulae of Gauss
2.4.7 GaussCodazzi Equations
2.4.8 Second-Order Differential Operator of Beltrami
2.4.9 Geodesic Curves in a Surface
2.4.10 Geodesic Torsion
2.5 Elliptic Equations for Grid Generation
2.5.1 Elliptic Grid Equations in Flat Spaces
2.5.1.1 Coordinate Transformation
2.5.1.2 Non-Steady Coordinates
2.5.1.3 Nonelliptic Grid Generation
2.5.2 Elliptic Grid Equations in Curved Surfaces
2.5.2.1 Transformation of the Surface Coordinates
2.5.2.2 The Fundamental Theorem of Surface Theory
2.5.2.3 Time-Dependent Surface Coordinates
2.5.2.4 Coordinate Generation Equations in a Hypersurface
2.6 Concluding Remarks
References
Chapter 3: Transfinite Interpolation (TFI) Generation Systems
3.1 Introduction
3.2 Grid Requirements
3.3 Transformations and Grids
3.4 Transfinite Interpolation (TFI)
3.4.1 Boolean Sum Formulation
3.4.2 Recursion Formulation
3.4.3 Blending Function Conditions
3.5 Practical Application of TFI
3.5.1 Linear TFI
3.5.2 Lagrangian TFI
3.5.3 Hermite Cubic TFI
3.6 Grid Spacing Control
3.6.1 Single-Exponential Function
3.6.2 Double-Exponential Function
3.6.3 Hyperbolic Tangent and Sine Control Functions
3.6.4 Arclength Control Functions
3.6.5 Boundary-Blended Control Functions
3.7 Conforming an Existing Grid to New Boundaries
3.8 Summary
References
Chapter 4: Elliptic Generation Systems
4.1 Introduction
4.2 Two-Dimensional Grid Generation
4.2.1 Harmonic Maps, Grid Control Maps and Poisson Systems
4.2.2 Discretization and Solution Method
4.2.3 Construction of Grid Control Maps
4.2.3.1 Laplace Grids
4.2.3.2 Arc Length Based Grids
4.2.3.3 Grid Orthogonality at the Boundary
4.2.3.4 Orthogonal Grids
4.2.3.5 Complete Grid Control at the Boundary
4.2.4 Best Practices
4.3 Surface Grid Generation
4.4 Volume Grid Generation
4.5 Research Issues and Summary
4.6 Further Information
References
Information Security Policies, Procedures, and Standards
Chapter 5: Hyperbolic Methods for Surface and Field Grid Generation
5.1 Introduction
5.2 Hyperbolic Field Grid Generation
5.2.1 Governing Equations for Hyperbolic Field Grid Generation
5.2.2 Numerical Solution of Hyperbolic Field Grid Generation Equations
5.2.3 Specification of Cell Sizes
5.2.4 Boundary Conditions
5.2.5 Grid Smoothing Mechanisms
5.3 Hyperbolic Surface Grid Generation
5.3.1 Governing Equations for Hyperbolic Surface Grid Generation
5.3.2 Numerical Solution of Hyperbolic Surface Grid Generation Equations
5.3.3 Communications with the Reference Surface
5.4 General Guidelines for High-Quality Grid Generation
5.4.1 Grid Stretching
5.4.2 Point Distribution Near Corners
5.5 Applications
5.5.1 Applications Using 2D Hyperbolic Field Grids
5.5.1.1 Three-Element Airfoil
5.5.1.2 Greater Antilles Islands and Gulf of Mexico
5.5.2 Applications Using 3D Hyperbolic Field Grids
5.5.2.1 SOFIA Telescope
5.5.2.2 Apache Helicopter
5.5.2.3 Space Shuttle Launch Vehicle
5.5.3 Applications Using Hyperbolic Surface Grids
5.5.3.1 Collar Grid
5.5.3.2 Pylon
5.5.3.3 V-22 Tiltrotor
5.5.3.4 X-CRV Crew Return Vehicle
5.6 Summary and Research Issues
Acknowledgments
References
Further Information
Chapter 6: Boundary Orthogonality in Elliptic Grid Generation
6.1 Introduction
6.2 Boundary Orthogonality for Planar Grids
6.2.1 Neumann Orthogonality
6.2.2 Dirichlet Orthogonality
6.2.2.1 Blending of Orthogonal and Initial Control Functions
6.3 Boundary Orthogonality for Surface Grids
6.3.1 Neumann Orthogonality
6.3.2 Dirichlet Orthogonality
6.4 Boundary Orthogonality for Volume Grids
6.4.1 Neumann Orthogonality
6.4.2 Dirichlet Orthogonality
6.5 Summary
References
Chapter 7: Orthogonal Generating Systems
7.1 Introduction
7.2 Generating Systems
7.2.1 Two-Dimensional Regions
7.2.1.1 Distortion Function and Boundary Conditions
7.2.1.2 Orthogonality Parameters
7.2.2 Curved Surfaces
7.2.2.1 Distortion Function and Boundary Conditions
7.2.2.2 Orthogonality Parameters
7.3 Numerical Solutions
7.3.1 Discretized Equations
7.3.2 Boundary Conditions
7.3.3 Convergence Criteria
7.3.4 Two-Dimensional Regions
7.3.4.1 Nonsymmetric Domains
7.3.4.2 Symmetric Domains
7.3.4.3 Domains with Nonorthogonal Boundaries
7.3.5 Curved Surfaces
7.3.5.1 Nonsymmetric Domains
7.3.5.2 Symmetric Domains
7.3.5.3 Domains with Nonorthogonal Boundaries
7.4 Summary
References
Further Information
Chapter 8: Harmonic Mappings
8.1 Introduction
8.2 Nondegenerate Planar Grids
8.2.1 Two-Dimensional Regular Grids
8.2.2 Discrete Analog of the Jacobian Positiveness
8.2.3 Irregular Two-Dimensional Meshes
8.3 Planar Harmonic Grid Generation
8.3.1 Problem Formulation
8.3.2 Variational Method for Irregular Planar Mesh Smoothing
8.4 Harmonic Maps Between Surfaces. Derivation of Governing Equations
8.4.1 Introductory Remarks
8.4.2 Theory of Harmonic Maps
8.4.3 Derivation of Governing Equations
8.5 Two-Dimensional Adaptive-Harmonic Structured Grids
8.5.1 Derivation of Equations
8.5.2 Numerical Implementation
8.6 Two-Dimensional Adaptive-Harmonic Irregular Meshes
8.6.1 Problem Formulation
8.6.2 Approximation of the Functional
8.6.3 Minimization of the Functional
8.6.4 Derivation of Computational Formulas
8.7 Adaptive-Harmonic Structured Surface Grid Generation
8.7.1 Derivation of Equations
8.7.2 Numerical Implementation
8.8 Irregular Surface Meshes
8.8.1 Problem Formulation
8.8.2 Approximation of the Functional
8.8.3 Minimization of the Functional
8.8.4 Derivation of Computational Formulas
8.9 Three-Dimensional Regular Grids
8.9.1 Derivation of Equations
8.9.2 Numerical Implementation
8.10 Three-Dimensional Irregular Meshes
8.10.1 Discrete Analog of the Jacobian Positiveness
8.10.2 Problem Formulation
8.10.3 Approximation of the Functional
8.10.4 Minimization of the Functional
8.10.5 Derivation of Computational Formulas
8.11 Results of Test Computations
8.11.1 Comparison Between the Winslow Method and the Variational Approach
8.11.2 Comparison Between the Finite-Difference Method for Two-Dimensional Adaptive-Harmonic Me...
8.11.3 Comparison Between the Finite-Difference Method for Adaptive- Harmonic Grid Generation o...
8.11.4 Comparison Between the Finite-Difference Method for Adaptive- Harmonic Three-Dimensional...
8.12 Conclusions
References
Chapter 9: Surface Grid Generation Systems
9.1 Introduction
9.2 Algebraic Surface Grid Generation
9.2.1 Distribution of Grid Points on the Boundary Curves
9.2.1.1 Hybrid Grid Density Functions
Algorithm 2.1 Hybrid Curve Point Distribution Algorithm
9.2.1.2 Determination of Weights ls,lk,l1 and Strengths ki
9.2.2 Interpolation of Grid Points Between Boundary Curves
9.2.3 NURBS Surface Grid Generation Examples
9.3 Elliptic Surface Grid Generation
9.3.1 Conformal Mapping on Surfaces
9.3.2 Formulation of the Elliptic Generator
9.3.3 Numerical Implementation
9.3.4 Control Function Computation
9.4 Summary and Research Issues
References
Chapter 10: A New Approach to Automated Multiblock Decomposition for Grid Generation: A Hypercube++ Approach
10.1 Introduction
10.2 Underlying Principles
10.2.1 NURBS Volume
10.2.2 Hypercube++ Structure
10.2.2.1 Hypercube and Its Limitations
10.2.2.2 Hypercube++ Structure
10.2.2.3 Data Structure
10.3 Best Practices
10.3.1 Hypercube++ Generation
10.3.2 Hypercube++ Merging
10.3.3 Main Features of Hypercube++ Approach
10.3.4 Applications
10.4 Research Issues and Summary
Further Information
References
Chapter 11: Composite Overset Structured Grids
11.1 Introduction
11.2 Domain Decomposition
11.2.1 Surface Geometry Decomposition
11.2.1.1 Seam Topologies
11.2.1.2 Block Topologies
11.2.2 Volume Geometry Decomposition
11.2.3 Chimera Hole-Cutting
11.2.3.1 Surface Normal Vector Test
11.2.3.2 Vector Intersection Test
11.2.3.3 Uniform Cartesian Test
11.2.4 Identification of Intergrid Boundary Points
11.3 Domain Connectivity
11.3.1 Donor Grid Identification
11.3.2 Donor Element Identification
11.3.2.1 Inside/Outside Test
11.3.2.2 Gradient Search
11.3.2.3 Spatial Partitioning
11.3.2.4 Combined Spatial Partitioning and Gradient Search
11.4 Research Issues
11.4.1 Surface Geometry Decomposition
11.4.2 Surface and Volume Grid Generation
11.4.3 Adaptive Refinement
11.4.4 Domain Connectivity
Defining Terms
Acknowledgment
Further Information
References
Chapter 12: Parallel Multiblock Structured Grids
12.1 Overview
12.2 Multiblock Grid Generation and Parallelization
12.3 Computational Aspects of Multiblock Grids
12.4 Description of the Standard Cube
12.5 Topology File Format for Multiblock Grids
12.6 Local Grid Clustering Using Clamp Technique
12.7 A Grid Generation Meta Language
12.7.1 Topology Input Language
12.8 Parallelization Strategy for Complex Geometry
12.8.1 Message Passing for Multiblock Grids
12.8.2 Parallel Machines and Computational Fluid Dynamics
12.9 Parallel Efficiency for Multiblock Codes
12.10 Parallel Solution Strategy: A Tangled Web
12.10.1 Parallel Numerical Strategy
12.10.2 Time Stepping Procedure
12.10.3 Parallel Solution Strategy
12.10.4 Solving Systems of Linear Equations: The CG Technique
12.10.5 Basic Description of GMRES
12.11 Future Work in Parallel Grid Generation and CFD
Acknowledgment
References
Chapter 13: Block-Structured Applications
13.1 Introduction
13.2 Guidelines for Generating Grids
13.2.1 Basic Decisions
13.2.2 Preparation for Grid Generation
13.2.2.1 Level of Detail
13.2.3 Getting Started
13.2.4 Generating the Grid
13.2.4.1 Controlling Grid Resolution
13.2.4.2 Overset Grid Methods
13.2.4.3 Spacing Normal to a Wall
13.2.4.4 Typical Distributions
13.2.5 Checking Quality
13.2.6 Grid Generation Example
13.2.7 Summary
13.2.7.1 When Is It Time To Change Codes?
13.3 CFD Application Study Guidelines
13.3.1 Managing Large CFD Studies
13.3.2 Modular Master Grid Approach
13.3.3 Communication
13.4 Grid Code Development Guidelines
13.4.1 Development Approach
13.4.2 Geometry Issues
13.4.3 Attention to Detail
13.5 Research Issues and Summary
13.6 Defining Terms
Further Information
References
Part II: Unstructured Grids
Introduction to Unstructured Grids
Chapter 14: Data Structures for Unstructured Mesh Generation
14.1 Introduction
14.2 Some Basic Data Structures
14.2.1 Linear Lists
14.2.1.1 Stacks and Queues
14.2.1.2 Sequential Allocation
14.2.1.3 Linked Allocation
14.2.2 A Simple Hash Table
14.3 Tree Structures
14.3.1 Binary Trees
14.3.1.1 How to Implement Binary Trees
14.3.2 Heaps
14.3.3 Binary Search Tree
14.3.4 Digital Trees
14.4 Multidimensional Search
14.4.1 Searching Point Data
14.4.2 Quadtrees
14.4.2.1 Region-Based Quadtrees
14.4.2.2 Point-Based Quadtrees
14.4.3 Binary Trees for Multidimensional Search
14.4.3.1 Bit Interleaving
14.4.4 Intersection Problems
14.4.4.1 Representing an Interval as a Point
14.5 Final Remarks
References
Chapter 15: Automatic Grid Generation Using Spatially Based Trees
15.1 Introduction
15.2 Recursive Domain Subdivision to Define Spatially Based Trees
15.3 Quadtrees and Octrees for Automatic Mesh Generation
15.4 Tree Construction for Automatic Mesh Generation
15.4.1 Preliminaries
15.4.2 Mesh Control and Octant Sizes
15.4.3 Definition of Octree
15.4.4 Information Stored in the Tree
15.5 Mesh Generation Within the Tree Cells
15.5.1 Meshing Interior Cells
15.5.2 Meshing Boundary Cells
15.5.2.1 Element Removal to Mesh Boundary Octants
15.5.2.2 Delaunay Point Insertion to Mesh Boundary Octants
15.5.2.3 Element Removal from a Pre-Triangulated Surface to Interior Octants
15.5.2.4 Hexahedral Element Creation from the Interior Octants to the Model Boundary
15.6 Mesh Finalization Processes
15.6.1 Node Point Repositioning
15.6.2 Elimination of Poorly Sized and Shaped Elements Caused by Interactions of the Object Bound...
15.6.3 Three-Dimensional Mesh Modifications to Improve Mesh Quality
15.6.4 A Couple of Examples
15.7 Closing Remarks
References
Chapter 16: DelaunayVorono Methods
16.1 Introduction
16.2 Underlying Principles
16.2.1 Vorono Diagram and Delaunay Triangulation
16.2.2 BowyerWatson Algorithm
16.2.3 TanemuraOgawaOgita Algorithm
16.2.4 Edge/Face Swapping
16.2.5 Grid Optimization
16.2.6 Constrained Delaunay Triangulation
16.3 Research Issues
References
Chapter 17: Advancing Front Grid Generation
17.1 Introduction
17.2 Mesh Generation Requirements
17.3 Geometry Modeling
17.3.1 Description of the Computational Domain
17.3.2 Curve and Surface Representation
17.3.3 The Advancing Front Technique
17.3.4 Front Updating
17.3.5 Characterization of the Mesh: Mesh Parameters
17.3.6 Mesh Control
17.3.7 Background Mesh
17.3.8 Distribution of Sources
17.3.9 Calculation of the Transformation T
17.3.10 Curve Discretization
17.3.11 Triangle Generation in Two-Dimensional Domains
17.3.12 Mesh Quality Enhancement
17.3.13 Surface Discretization
17.3.14 Generation of Tetrahedra
17.3.15 Mesh Quality Assessment
17.4 Data Structures
17.4.1 The Alternating Digital Tree
17.4.2 Geometric Searching
17.4.3 Geometric Intersection
17.4.4 The Use of the ADT for Mesh Generation
17.5 Conclusions
References
Chapter 18: Unstructured Grid Generation Using Automatic Point Insertion and Local Reconnection
18.1 Introduction
18.2 Unstructured Grid Generation Procedure
18.3 Two-Dimensional Application Examples
18.3.1 Multi-Element Airfoil
18.3.2 Mediterranean Sea
18.4 Three-Dimensional Surface Grid Generation
18.4.1 Edge Grid Generation Procedure
18.4.2 Surface Grid Generation Procedure
18.5 Three-Dimensional Surface Grid Generation Application Examples
18.5.1 Generic Shell
18.5.2 Hawaiian Islands
18.6 Surface and Volume Grid Generation Best Practice
18.7 Three-Dimensional Application Examples
18.7.1 Pump Cover
18.7.2 SUV Interior
18.7.3 NASA Space Shuttle Orbiter
18.7.4 Launch Vehicle
18.7.5 Destroyer Hull
18.8 Summary
Acknowledgments
References
Chapter 19: Surface Grid Generation
19.1 Introduction
19.2 Surface modeling
19.2.1 Geometrical Definition
19.2.1.1 Curves
19.2.1.2 Surfaces
19.2.2 Topological Description
19.3 Surface Discretization
19.3.1 Grid Control Function
19.3.2 Grid Quality
19.4 Triangulation of Surfaces
19.4.1 Grid Generation Procedure
19.4.2 Computation of the Local Coordinates of the Edge Endpoints
19.4.3 Curve Discretization
19.4.3.1 Discretization Using a Distribution Function
19.4.3.2 Discretization Using the Mapping T
19.4.4 Computation of Coordinates in the Parameter Plane
19.4.5 Orientation of Initial Front
19.4.6 Grid Generation in the Parameter Plane
19.4.6.1 The Modified 2D AFT
19.4.6.2 Grid Characteristics in the Parameter Plane
19.4.6.3 Influence of the Surface Parametrization
19.4.7 Finding the Location of the Ideal Point
19.4.8 Surface Grid Enhancement Techniques
19.4.8.1 Diagonal Swapping
19.4.8.2 Grid Smoothing
19.5 Orientation of the Assembled Surface
Further Information
References
Chapter 20: Nonisotropic Grids
20.1 Introduction
20.2 The Classical Delaunay Mesh Generation Method
20.2.1 Scheme of a Classical Mesh Generator
20.2.2 Boundary Mesh Creation
20.2.2.1 The Delaunay Kernel
20.2.2.2 Meshing the Enclosing Rectangle
20.2.3 Creating the Mesh of a Domain V
20.3 Scheme of an Anisotropic Mesh Generator
20.3.1 The Mesh of the Domain
20.4 Fundamental Definitions
20.4.1 Metric at a Point
20.4.2 Length of a Segment
20.5 The Anisotropic Delaunay Kernel
20.5.1 The Delaunay Measure
20.5.2 Approach Using Only One Metric
20.5.3 Approach Using Two Metrics
20.5.4 Approach Using Four Metrics
20.6 The Field Points Definition
20.6.1 The Control Space
20.6.1.1 Geometrical Interpretation of the Metrics
20.6.1.2 Interpolation on a Segment
20.6.1.3 Interpolation over a Triangle
20.6.2 Computation of the Edge Length
20.6.3 Field Point Creation
20.6.4 Filtration of the Field Points
20.6.5 Insertion of the Field Points
20.7 Optimization
20.7.1 Element Quality
20.7.2 Diagonal Swapping
20.7.3 Point Relocation
20.7.3.1 Relocation with Unit Length
20.7.3.2 Relocation with Optimal Shape
20.8 Metric Construction
20.8.1 Computation of the Hessian
20.8.2 Remark on Metric Computation
20.8.3 Metric Associated with Classical Norms
20.8.4 Metric with Relative Error
20.8.5 Metric Intersection
20.9 Loop of Adaptation
20.10 Application Examples
20.10.1 NavierStokes Solver
20.10.2 Flow Over a Backward Step
20.10.3 Transonic Turbulent Flow Over a RAE2822
20.11 Application to Surface Meshing
20.12 Concluding Remarks
Acknowledgment
References
Chapter 21: Quadrilateral and Hexahedral Element Meshes
21.1 Introduction
21.2 Block-Decomposition Methods
21.3 Superposition Methods
21.4 The Spatial Twist Continuum
21.5 Other Approaches
21.6 Software and Online Information
Acknowledgment
References
Chapter 22: Adaptive Cartesian Mesh Generation
22.1 Introduction
22.2 Overview of Cartesian Grids
22.2.1 Geometric Requirements of Cartesian Finite Volume Flow Solvers
22.2.2 Data Structures
22.2.3 Surface Geometry
22.2.3.1 Triangle Intersections
22.2.3.2 Constrained Retriangulation
22.2.3.3 Inside/Outside
22.2.3.4 Automatic Treatment of Degeneracies
22.3 Cartesian Volume Mesh Generation
22.3.1 Overview
22.3.2 Volume Mesh Generation
22.3.2.1 Initial Mesh Specification and Integer Coordinates
22.3.2.2 Efficient Spatial Searches
22.3.2.3 Inside/OutsideDetermination
22.3.2.4 Neighborhood Traversal
22.3.3 Cell Subdivision and Mesh Adaptation
22.3.4 Body Intersecting Cells
22.3.4.1 Rapid Intersection with Coordinate Aligned Regions
22.3.4.2 Polygon Clipping
22.4 Examples
22.4.1 Steady State Simulations
22.4.1.1 ONERA M6
22.4.1.2 Examples with Complex Geometry
22.4.1.3 Transport Aircraft with High-Lift System Deployed
22.5 Research Issues
22.5.1 Moving Geometry
22.5.2 NURBS Surface Definitions
22.5.3 Viscous Applications
22.6 Summary
Appendix 1: Integer Numbering of Adaptive Cartesian Meshes
Acknowledgment
References
Chapter 23: Hybrid Grids
23.1 Introduction
23.2 Underlying Principles
23.2.1 Historical Review
23.2.2 The Trend from Unstructured to Hybrid Grids
23.2.3 The Trend from Structured to Hybrid Grids
23.2.4 Potential Computational Benefits of Using Hybrid Meshes
23.3 Best Practices
23.3.1 Mesh Generation Techniques Employed in the SAUNA System
23.3.1.1 Overview of Development
23.3.1.2 Structured Grid Generation
23.3.1.3 Semistructured Grid Generation
23.3.1.4 Unstructured Grid Generation
23.3.2 Interfacing Different Grid Types
23.3.2.1 Interfacing Structured and Semistructured Grids
23.3.2.2 Interfacing Structured and Unstructured Grids
23.3.2.3 Interfacing Semistructured and Unstructured Grids
23.3.3 Data Structures for Describing Hybrid Grids
23.3.4 Examples of Hybrid Meshes
23.3.4.1 Creation of a Block-Structured/Unstructured Grid for a Civil Aircraft
23.3.4.2 Creation of a Semistructured/Unstructured Grid for a Submarine
23.3.4.3 Creation of a General Hybrid Grid for a Store Below a Research Aircraft
23.4 Research Issues and Summary
Acknowledgments
References
Chapter 24: Parallel Unstructured Grid Generation
24.1 Introduction
24.2 Requirements for Parallel Mesh Generation
24.3 Classification of Parallel Mesh Generators
24.4 Meshing Interfaces Along with subdomains
24.5 Premeshing Interfaces
24.4.1 Initial Coarse Mesh Partitioning
24.4.2 Tree Partitioning
24.4.3 Prepartitioning
24.6 Postmeshing Interfaces
24.7 Conclusion
References
Chapter 25: Hybrid Grids and Their Applications
25.1 Introduction
25.2 Underlying Principles
25.2.1 The Structured Marching Method for Prisms
25.2.1.1 Determination of the Marching Vectors
25.2.1.2 Marching Step Size
25.2.1.3 Smoothing Steps
25.2.1.4 Constraints Imposed to Enhance Quality
25.2.1.5 Automatic Adjustment of the Prism Layer Thickness
25.2.2 The Octree-Advancing Front Methods for Tetrahedra
25.2.2.1 Length Scales
25.2.2.2 Octree Guides Advancing Front Mesh Generation
25.2.2.3 Anisotropic Surface Meshes
25.2.2.4 Automatic Partial Remeshing
25.3 Best Practices
25.3.1 High Speed Civil Transport (HSCT) Aircraft
25.3.2 Adapted Hybrid Mesh
25.3.3 Resolution of Multiple Wakes
25.3.4 Deforming Hybrid Mesh in 2D
25.3.5 Turbomachinery Blade with Tip Clearance
25.3.6 ABB Burner Case
25.4 Research Issues and Summary
Further Information
References
Chapter 26: Unstructured Grids: Procedures and Applications
26.1 Introduction
26.2 Grids Constructed by Delaunay Triangulation The General Procedure
26.3 Unstructured Grid Control Using a Background Grid and Sources
26.4 Unstructured Grids of Triangles
26.5 Hybrid Grids of Quadrilaterals and Triangles
26.6 Unstructured Tetrahedral Grids
26.6.1 Dassault Falcon
26.6.2 THRUST Supersonic Car
26.7 Non-Isotropic Grid Generation for Viscous Flow Simulation
26.8 Parallel Unstructured Grid Generation
26.9 Summary
Appendix: Graphics User Interfaces
Acknowledgment
References
Part III: Surface Definition
Introduction to Surface Definition
Chapter 27: Spline Geometry: A Numerical Analysis View
27.1 Background and Introduction
27.2 A Functional Approach to Splines
27.3 Basics of Spline Theory
27.4 B-Splines
27.4.1 Description and Examples of B-Splines
27.4.2 Evaluation
27.4.3 Robustness of the B-Spline Representation
27.4.4 A Representation Format for Univariate Splines
27.5 Approximation with Splines
27.6 Constructing Spline Functions
27.6.1 Least Squares Approximation
27.6.2 Interpolation Methods
27.6.2.1 General Interpolation
27.6.2.2 Underdetermined Problems Knots at the Data Abscissae
Method 1: Natural Spline Interpolation
Method 2: Complete Spline Interpolation
Method 3: Not-a-knot Interpolation
Method 4: Even Degree Interpolation
27.7 Parametric Curves and Rational Splines
27.7.1 Parametric Curves
27.7.2 Rational Splines
27.7.3 Representation of Rational Splines and an Example
27.8 Surfaces
27.8.1 Tensor Product Splines
27.8.2 Interpolation and Approximation on a Rectangular Grid
27.8.3 Interpolation and Approximation of Scattered Data
27.8.4 Construction of Parametric Spline Surfaces from Rectangular Data
27.8.5 Other Methods of Construction of Surfaces
27.9 Functional Composition
References
Chapter 28: Computer-Aided Geometric Design
28.1 History
28.2 Basic Principles
28.3 Bzier Curves
28.4 Cubic Hermite Curves
28.5 B-Splines
28.6 Cubic Interpolation and Approximation
28.7 Bzier Patches
28.8 Composite Surfaces
28.9 Rational Curves and Surfaces NURBS
References
Chapter 29: Computer-Aided Geometric Design Techniques for Surface Grid Generation
29.1 Introduction
29.1.1 Surface Refinement and Reparametrization
29.1.2 Approximation of Discontinuous Geometries
29.1.3 SurfaceSurface Intersection
29.2 Surface Refinement and Reparametrization Underlying Principles and Best Practices
29.2.1 Approaches to Solving the Problem
29.2.2 Modifying the Existing Surface
29.2.3 The Surface Approximation Scheme
29.2.4 Boundary Curve Approximation
29.2.5 Finding an Interpolating Surface
29.2.6 Finding Interior Interpolation Points
29.3 Approximation of Discontinuous Geometries Underlying Principles and Practices
29.3.1 The Algorithm and References
29.3.2 Computing the Initial Coons Patch
29.3.3 Projecting the Coons Patch onto the Original Surfaces
29.3.4 Computing Additional Approximation Conditions
29.3.5 Constructing a Local Surface Approximant
29.3.6 Error Estimation
29.3.7 Connecting the Local B-Spline Approximants
29.3.8 Examples
29.4 SurfaceSurface Intersection Underlying Principles and Best Practices
29.4.1 The Intersection Algorithm.
29.4.2 Triangulation
29.4.3 Triangle Intersection
29.4.4 Intersection Preprocessing Using a Tree Structure
29.4.5 Data Structure, Loop Detection, and Curve Tracing
29.4.6 Refinement
29.5 Research Issues and Summary
29.5.1 Surface Refinement and Reparametrization
29.5.2 Approximation of Discontinuous Geometries
29.5.3 SurfaceSurface Intersection
Acknowledgments
Further Information
References
Chapter 30: NURBS in Structured Grid Generation
30.1 Introduction
30.2 NURBS Formulation
30.3 Transforming and Generating Procedures
30.3.1 General Circular Arc to NURBS Curve
30.3.2 Conic Arc to NURBS Curve
30.3.3 Cubic Parametric Curve to NURBS Curve
30.3.4 Composite Curve to NURBS Curve
30.3.5 Superellipse to NURBS Curve
30.3.6 Bicubic Parametric Spline Surface to NURBS Surface
30.3.7 Surface of Revolution to NURBS Surface
30.3.8 Transfinite Interpolation for NURBS Surface
30.3.9 Cascading Technique for NURBS Surface
30.4 Grid Redistribution
30.4.1 Reparametrization Algorithm
30.4.2 Singularity Control
30.5 Volume Grid Generation by NURBS Control Volume
30.5.1 Ruled Volume
30.5.2 An Extruded Volume
30.5.3 Volume of Revolution
30.5.4 Composite Volume
30.5.5 Transfinite Interpolation Volume
30.6 Conclusion and Summary
Acknowledgment
References
Chapter 31: NASA IGES and NASA-IGES NURBS Only Standard
31.1 Introduction
31.1.1 Purpose
31.1.2 Scope
31.1.3 Background
31.1.4 NASA Support
31.2 Underlying Principles (the CFD Process)
31.2.1 The CFD Analysis Process
31.2.2 The CFD Design Process
31.2.3 Problems with Pre-NASA-IGES Methods
31.2.4 CFD Design Utilizing NASA-IGES
31.2.5 CFD Design Utilizing the Supplied Database Information Format
31.2.6 General Information on Data Description
31.2.6.1 Entity Description Overview
31.2.6.2 Coordinate System
31.2.6.3 Common Information
31.3 Best Practices
31.3.1 Multidisciplinary Data Exchange Standards
31.3.2 Summary of Entity Types and Recommended Usage
Entity 0 : Null Entity
Entity 100: Circular Arc
Entity 102: Composite Curve
Entity 104: Conic Arc
Entity 106: Copious Data
Entity 110: Line
Entity 116: Point
Entity 124: Transformation Matrix
Entity 126: Rational B-Spline Curve
Entity 128: Rational B-Spline Surface
Entity 141: Boundary
Entity 142: Curve on a Parametric Surface
Entity 143: Bounded Surface
Entity 212: General Note
Entity 308: Subfigure Definition
Entity 314: Color Definition
Entity 402: Associativity Instance
Entity 406: Property, Form15: Name
Entity 408: Singular Subfigure Instance
31.3.3 Case Studies:
31.3.3.1 Blade Surface Geometry Modeling
31.3.3.2 Computational Fluid Dynamics
31.3.3.3 Multidisciplinary Geometry, Grid, and Analysis Association
31.3.4 Other NASA-IGES Compatible Software
31.4 Research Issues and Summary
Further Information
References
Part IV: Adaptation and Quality
Introduction to Adaptation and Quality
Chapter 32: Truncation Error on Structured Grids
32.1 Introduction
32.2 Order on Nonuniform Spacing
32.2.1 Order with Fixed Distribution Function
32.2.2 Order with Fixed Number of Points
32.3 Effect of Numerical Metric Coefficients
32.4 Evaluation of Distribution Functions
32.5 Two-Dimensional Forms
References
Chapter 33: Grid Optimization Methods for Quality Improvement and Adaptation
33.1 Introduction
33.1.1 Notation and General Framework of the Chapter
33.2 RegularityOrthogonality Formulation
33.2.1 Measure of the Orthogonality
33.2.2 Measure Of The Regularity
33.2.3 Global Functional
33.2.4 Origin of the RegularityOrthogonality Functional
33.2.5 Discussion of the RegularityOrthogonality Functional
33.3 Deformation Formulation
33.3.1 Measure of the Cell Deformation
33.3.2 Characterization of Functional Sigma
33.3.3 Mechanical Interpretation of the Method
33.3.4 Cell Deformation and Measure of the Mesh Quality
33.3.5 Choice of the Functionals in Two and Three Dimensions
33.3.5.1 Two-Dimensional Functional
33.3.5.2 Surface Functional
33.3.5.3 ThreeDimensional Functional
33.4 Handling of an Initial Grid
33.4.1 General Principle
33.4.2 RegularityOrthogonality Formulation
33.4.3 Deformation Formulation
33.4.3.1 Reference Configurations
33.4.3.2 Toward Conformity and Exact Orthogonality in 2D
33.4.4 Summary of the Optimization
33.5 Handling of Adaptation
33.5.1 Introduction
33.5.2 General Principle
33.5.3 Use of Error Indicators
33.5.4 Use Of Error Estimators
33.5.5 Formulation Using Volume Integral
33.6 Optimization Algorithm
33.6.1 General Algorithm
33.6.2 Handling of Conditions on the Boundary
33.6.3 Handling of Multidomain Topologies
33.7 Extension to Unstructured Meshes
33.7.1 Regularity Criterion
33.7.2 Deformation Formulation
33.7.3 Adaption
33.8 Summary and Research Issues
Appendix A
Appendix B
References
Chapter 34: Dynamic Grid Adaption and Grid Quality
34.1 Introduction
34.2 Problem Statement
34.3 Theory and Principles
34.3.1 Fundamentals
34.3.2 Adaptive Algorithm Implementations (DSAGA, SIERRA)
34.3.2.1 DSAGA
34.4 Grid Quality
34.5 SIERRA
34.5.1 Weight Function
34.5.2 Transformation to Physical Space
34.5.3 Grid Adaptation Cut-Off Criteria
34.5.4 Interim Steps
34.6 Results
34.6.1 Experimental Comparisons
34.6.1.1 Hypersonic 2D Compression Corner
34.6.1.2 Supersonic 3D Symmetric Corner Flow
34.7 Summary and Conclusions
34.8 Research Issues, Current and Future
References
NCSU Adaptive Grid Bibliography
Chapter 35: Grid Control and Adaptation
35.1 Introduction
35.2 Unstructured Mesh Control
35.2.1 Characterization of an Unstructured Mesh
35.2.2 Advancing Front Grid Control
35.2.2.1 The Background Mesh
35.2.2.2 Sources
35.2.3 Delaunay Grid Control
35.2.3.1 Automatic Point Creation Driven by the Boundary Point Distribution
35.2.3.2 Automatic Point Creation Controlled by a Background Mesh
35.2.3.3 Automatic Point Creation by the Use of Sources
35.3 Mesh Quality Enhancement
35.3.1 Mesh Cosmetics
35.3.1.1 Edge Swapping
35.3.1.2 Nodal Reconnection
35.3.1.3 Edge Deletion
35.3.1.4 Spatial Smoothing
35.3.2 Grid Quality Statistics
35.4 Mesh Adaption
35.4.1 Introduction
35.4.2 Error Indicator in 1D
35.4.3 Extension to Multidimensions
35.4.4 Mesh Enrichment
35.4.5 Mesh Movement
35.4.6 Adaptive Remeshing
35.4.7 Grid Adaptation Using the Delaunay Triangulation with Sources
35.4.7.1 Surface Adaptation
35.4.7.2 Field Adaptation
References
Chapter 36: Variational Methods of Construction of Optimal Grids
36.1 Introduction
36.2 Constructions of the Functionals Formalizing the Optimality Criteria
36.2.1 Analysis of the Functionals (U) and (A) in One-Dimensional Case
36.2.2 Construction of Two-Dimensional and Three-Dimensional Functionals (U), (O), (A)
36.2.3 Boundary Conditions. The Analysis of Boundary Value Problems in the Two-Dimensional Case
36.3 Effective Algorithms of Optimal Grid Generation
36.3.1 Organization of the Iterative Process
36.3.2 Multiply Connected Optimal Grids in Two-Dimensional Domains. The Program MOPS-2a
36.3.2.1 Initial Approximation of Grid
36.3.2.2 Automatic Overlapping of Subdomains
36.3.3 Algorithm of Two-Dimensional Optimal Adaptive Grid Generation. The Program LADA
36.3.3.1 Set of Points for Minimization of the Functional J
36.3.3.2 Organization of Calculations
36.3.3.3 About the Choice of Control Parameters
36.4 Simulation of Rotational Flows of Gas in Channels of Complex Geometries by Means of Optimal ...
36.5 Conclusion
References
Chapter 37: Moving Grid Techniques
37.1 Introduction
37.2 Underlying Principles
37.2.1 Transformation of Variables
37.2.2 The Method of Characteristics (MoC)
37.2.3 Equidistribution
37.3 Best Practices
37.3.1 Moving Finite Differences (MFD)
37.3.2 Moving Finite Elements (MFE)
37.3.2.1 Some Properties of the Moving Grid for MFE
37.3.3 Related Approaches
37.3.3.1 The Deformation Method
37.3.3.2 Other Techniques
37.4 Research Issues and Summary
Further Information
References
Appendix A: Grid Software
Appendix B: Grid Configurations
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